Chebyshev series: Derivation and evaluation

In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.


Theoretical background
If H(x, y, t) can be expanded in powers of t in the form Hðx; y; tÞ ¼ where h n is independent of x and y, and f n (x) and g n (x) are different functions, we utilise the language employed by Rainville see p.170 in [1] and call H(x, y, t) a bilateral generating function. A double Chebyshev series is one that has two univariate Chebyshev polynomials T n,p (α, β) = T n (α)T p (β) as products in each term of the series, where T n (α) = cos(nθ) and α = cos(θ). A contour integral method used by Brafman [2] was used to obtain generating functions satisfying a Rodigues-type formula reducible to the form where a and b are constants, not both zero and F(x) is independent of n and differentiable an arbitrary number of times.

Recent developments
Mathematicians around the world have remained interested in generating functions from 1965 to present. Publications from India, Japan, Canada, the United States, England, Russia, Germany, and other European nations provide proof of this interest. Three excellent works on special functions were published in 1968 alone. Each of these took into account the idea of the generating function from a group-theoretic perspective. They were authored by American Willard Miller, Jr. [3], Canadian James D. Talman [4], and Russian N.J. Vilenkin [5]. (V. N. Singh translated Vileqkin's novel into English). Publications on generating functions by N. A. AI-Salam and W. A. AI-Salam [6], J. W. Brown [7], and L. Carlitz [8] are only a few of the many intriguing ones along with more recent interesting publications. The Special Functions Section of the Mathematical Reviews contains a wealth of references to publications on generating functions. Literature on Chebyshev polynomials are presented in great detail in section 18 in [9]. These polynomials have well known generating functions in current literature and are used widely in all areas of mathematics and science. A particular popular application of these generating functions is in the solution of partial differential equations. Famous mathematicians like Lanczos [10] used the bivariate form of these generating functions along with their strong convergence properties in solving ordinary differential equations. In the work by Mason [11] the bivariate form of Chebyshev polynomials was employed in studying polynomial approximation. Examples of bilateral generating functions and their derivations are detailed in chapter 1 in McBride [12] and in the work by Mohammad [13]. The Chebyshev polynomial has also been studied and used in numerical solutions of initial boundary equations [14][15][16]. Generating functions involving special functions have been studied in the work by Meena et al. [17][18][19][20]. Solutions of partial differential equations in terms of two-dimensional Cbebyshev series have been studied with possible applications to the eigenvalue problem for a vibrating L-shaped membrane. In this work we provide a closed form solution to the bilateral generating function featuring the double summation of Chebyshev polynomials expressed in terms of the incomplete gamma function similar to the forms given in the works [12,13]. In section 3, the Chebyshev contour integral formula is derived. In section 4, we give a detailed account of the incomplete gamma including integral and summation definitions. In section 5, the contour integral representations for the incomplete gamma function are derived. section 6, we formulate the main theorem and deduce a few propositions and examples in terms of constant and composite functions. In section 7, we look at the limiting case of the difference of the bivariate Chebyshev polynomials expressed in terms of the incomplete gamma function. Our preliminaries start with the contour integral method [21], applied to the formula for the Chebyshev generating function given by equation (18.12.8) in [9]. Let a, α, β, k and w be general complex numbers, n 2 [0, 1) and p 2 [0, 1), where the contour integral form of the Chebyshev generating function is given by T n ðaÞT p ðbÞa w w À kþnþpÀ 1 dw where |Re(w)| < 1. We will use Eq (3) to derive equivalent sums for the left-hand side and a Special function form for the right-hand side. The derivation of the summation follows the method used by us in [21] which involves Cauchy's integral formula. The generalized Cauchy's integral formula is given by where C is in general, an open contour in the complex plane where the bilinear concomitant has the same value at the end points of the contour. This method involves using a form of Eq (4) then multiply both sides by a function, then take a definite sum of both sides. This yields a definite sum in terms of a contour integral. A second contour integral is derived by multiplying Eq (4) by a function and performing some substitutions so that the contour integrals are the same.

The left-hand side contour integral
In this section we derive the infinite sum representation involving the product of two generalized Chebyshev polynomials over independent indices for the left-hand side of Eq (3). Using a generalization of Cauchy's integral formula (4), first replace y ! log(a), k ! k − n − p then multiply both sides by T n (α)T p (β) and take the sums over n 2 [0, 1) and p 2 [0, 1) and simplify to get X n;p�0 T n ðaÞT p ðbÞlog kÀ nÀ p ðaÞ Gðk À n À p þ 1Þ from Eq (3) where |Re(w)| < 1 and Im(w) > 0 in order for the sums to converge. Apply Tonelli's theorem for multiple sums, see page 177 in [22] as the summands are of bounded measure over the space C � ½0; 1Þ � ½0; 1Þ.

Derivation of the incomplete gamma function contour integral representations
In this section we derive the general case of the Incomplete Gamma function in terms of the Cauchy contour integral. This formula will be used in the proceeding section to derive the equivalent Incomplete Gamma function contour integral representations for the right-hand side of Eq (3). Using a generalization of Cauchy's integral formula (4), first replace y ! y + log (a) then multiply both sides by e xy and take the definite integral over y 2 [0, 1) and simplify to get from equation (3.462.12) in [25] where Re(x) < 0, |arglog(a)| < π.

The limiting case of the difference of a negative index
In this section we will derive a few generating functions using the identity T n (−x) = −1 n T n (x) which is listed in Table (18.6.1) in [9]. We proceed by using Eq (27) and forming a second equation by replacing α ! α, β ! −β taking their difference and simplifying. Next we evaluate five cases listed below when α = β = 1, α = β = 2, α = β = 3, α = β = 4, α = β = 5. In order to simplify the right-hand sides of these formulae we apply the limit and simplify. The simplification process is not very easy and tedious, however the results are very inetersting.

Generating functions involving the Chebyshev polynomial
In this section we derive a few exponential generating functions involving the Chebyshev polynomial. The method involves simultaneous equations and ordinary differential equations.